## [1] "es_ES.UTF-8/es_ES.UTF-8/es_ES.UTF-8/C/es_ES.UTF-8/en_US.UTF-8"

1. [20 puntos] El archivo covid19_CR.csv contiene la cantidad de casos nuevos diarios registrados en Costa Rica desde el 6 de marzo del 2020 hasta el 23 de abril de 2021. Con la tabla realice lo siguiente:

'data.frame':   414 obs. of  2 variables:
 $ FECHA       : chr  "2020-03-06" "2020-03-07" "2020-03-08" "2020-03-09" ...
 $ casos_nuevos: int  2 5 3 2 1 9 1 3 1 8 ...

a) Convierta a serie de tiempo en un objeto tipo ts utilizando una frecuencia de forma que el Patrón-Estacional sea semanal.

Time Series:
Start = c(1, 6) 
End = c(60, 6) 
Frequency = 7 
 [1]  2  5  3  2  1  9  1  3  1  8  6  9 19 18 26  4 17 24 19 24 30 32 32 19 16
[26] 17 28 21 20 19 19 13 16 19 37 19 19 18 17  6  8 16  7  6  5  2  7 12  5  1
[51]  6  2  2  8  8  6  6  8  6  3 13  6  4  8  7 12  9  3 11 15 13 10 10  3 16
 [ reached getOption("max.print") -- omitted 339 entries ]

(b) Utilizando el último mes de abril para pruebas y el resto de fechas para entrenamiento genere los modelos de HOLT-WINTERS, HOLT-WINTERS Calibrado y Redes Neuronales, luego en un solo gráfico muestre la serie de entrenamiento, la serie de prueba y el resultado de la predicción de cada uno de los modelos anteriores.

HoltWinters

         Point Forecast     Lo 80    Hi 80     Lo 95    Hi 95
57.57143       559.6448 411.04690 708.2426 332.38393 786.9056
57.71429       519.7350 367.15562 672.3144 286.38497 753.0850
57.85714       525.4608 368.79858 682.1230 285.86661 765.0550
58.00000       312.7390 151.89525 473.5828  66.74970 558.7284
58.14286       265.0959  99.97441 430.2174  12.56437 517.6274
58.28571       502.9149 333.42202 672.4078 243.69790 762.1319
58.42857       554.6465 380.69088 728.6022 288.60432 820.6887
58.57143       551.4222 356.06848 746.7759 252.65446 850.1899
58.71429       511.5124 311.91110 711.1137 206.24855 816.7763
58.85714       517.2382 313.29415 721.1823 205.33267 829.1438
59.00000       304.5165  96.13645 512.8965 -14.17326 623.2062
59.14286       256.8733  43.96623 469.7804 -68.73997 582.4866
59.28571       494.6923 277.16895 712.2157 162.01902 827.3657
59.42857       546.4240 324.19689 768.6510 206.55700 886.2909
59.57143       543.1996 301.42933 784.9699 173.44386 912.9554
 [ reached 'max' / getOption("max.print") -- omitted 8 rows ]
         Point Forecast     Lo 80    Hi 80     Lo 95    Hi 95
57.57143       559.6448 411.04690 708.2426 332.38393 786.9056
57.71429       519.7350 367.15562 672.3144 286.38497 753.0850
57.85714       525.4608 368.79858 682.1230 285.86661 765.0550
58.00000       312.7390 151.89525 473.5828  66.74970 558.7284
58.14286       265.0959  99.97441 430.2174  12.56437 517.6274
58.28571       502.9149 333.42202 672.4078 243.69790 762.1319
58.42857       554.6465 380.69088 728.6022 288.60432 820.6887
58.57143       551.4222 356.06848 746.7759 252.65446 850.1899
58.71429       511.5124 311.91110 711.1137 206.24855 816.7763
58.85714       517.2382 313.29415 721.1823 205.33267 829.1438
59.00000       304.5165  96.13645 512.8965 -14.17326 623.2062
59.14286       256.8733  43.96623 469.7804 -68.73997 582.4866
59.28571       494.6923 277.16895 712.2157 162.01902 827.3657
59.42857       546.4240 324.19689 768.6510 206.55700 886.2909
59.57143       543.1996 301.42933 784.9699 173.44386 912.9554
 [ reached 'max' / getOption("max.print") -- omitted 8 rows ]
[1] "Model: ETS(A,A,A)"
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[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
[1] "Model: ETS(A,A,A)"
HoltWinters(y = datos, alpha = 0.4, beta = 0.3, gamma = 0.1)
         Point Forecast        Lo 80     Hi 80      Lo 95     Hi 95
57.57143       585.3225   381.026468  789.6185   272.8787  897.7662
57.71429       638.3513   388.976271  887.7264   256.9651 1019.7376
57.85714       669.7678   347.394187  992.1415   176.7399 1162.7958
58.00000       482.9130    65.228790  900.5971  -155.8798 1121.7058
58.14286       410.1700  -120.213309  940.5532  -400.9813 1221.3212
58.28571       650.8701    -6.379086 1308.1192  -354.3058 1656.0459
58.42857       755.0108   -41.219632 1551.2412  -462.7185 1972.7401
58.57143       785.8356  -171.308027 1742.9793  -677.9893 2249.6606
58.71429       838.8645  -276.187504 1953.9165  -866.4604 2544.1894
58.85714       870.2810  -412.073131 2152.6352 -1090.9104 2831.4724
59.00000       683.4261  -774.966899 2141.8191 -1546.9935 2913.8458
59.14286       610.6831 -1031.963445 2253.3297 -1901.5280 3122.8943
59.28571       851.3833  -983.304048 2686.0705 -1954.5288 3657.2953
59.42857       955.5239 -1078.632386 2989.6803 -2155.4497 4066.4976
59.57143       986.3488 -1263.042021 3235.7396 -2453.7976 4426.4952
 [ reached 'max' / getOption("max.print") -- omitted 8 rows ]
         Point Forecast        Lo 80     Hi 80      Lo 95     Hi 95
57.57143       585.3225   381.026468  789.6185   272.8787  897.7662
57.71429       638.3513   388.976271  887.7264   256.9651 1019.7376
57.85714       669.7678   347.394187  992.1415   176.7399 1162.7958
58.00000       482.9130    65.228790  900.5971  -155.8798 1121.7058
58.14286       410.1700  -120.213309  940.5532  -400.9813 1221.3212
58.28571       650.8701    -6.379086 1308.1192  -354.3058 1656.0459
58.42857       755.0108   -41.219632 1551.2412  -462.7185 1972.7401
58.57143       785.8356  -171.308027 1742.9793  -677.9893 2249.6606
58.71429       838.8645  -276.187504 1953.9165  -866.4604 2544.1894
58.85714       870.2810  -412.073131 2152.6352 -1090.9104 2831.4724
59.00000       683.4261  -774.966899 2141.8191 -1546.9935 2913.8458
59.14286       610.6831 -1031.963445 2253.3297 -1901.5280 3122.8943
59.28571       851.3833  -983.304048 2686.0705 -1954.5288 3657.2953
59.42857       955.5239 -1078.632386 2989.6803 -2155.4497 4066.4976
59.57143       986.3488 -1263.042021 3235.7396 -2453.7976 4426.4952
 [ reached 'max' / getOption("max.print") -- omitted 8 rows ]

Redes Neuronales

Series: train.ts 
Model:  NNAR(22,1,40)[7] 
Call:   nnetar(y = train.ts, size = 40)

Average of 20 networks, each of which is
a 22-40-1 network with 961 weights
options were - linear output units 

sigma^2 estimated as 97.17
      Point Forecast
57.57       649.2158
57.71       712.5163
57.86       506.9182
58.00       424.1000
58.14       341.1344
58.29       474.8286
58.43       579.1668
58.57       625.3498
58.71       603.6568
58.86       535.3503
59.00       452.2606
59.14       408.6931
59.29       541.1187
59.43       619.4778
59.57       690.0131
59.71       667.1313
59.86       542.8352
60.00       511.9385
60.14       497.6412
60.29       560.1245
60.43       768.5957
60.57       763.6815
60.71       804.0039
Time Series:
Start = c(1, 6) 
End = c(60, 6) 
Frequency = 7 
          train.ts Original       HW HW Calibrado Redes neuronales
 1.714286        2       NA       NA           NA               NA
 1.857143        5       NA       NA           NA               NA
 2.000000        3       NA       NA           NA               NA
 2.142857        2       NA       NA           NA               NA
 2.285714        1       NA       NA           NA               NA
 2.428571        9       NA       NA           NA               NA
 2.571429        1       NA       NA           NA               NA
 2.714286        3       NA       NA           NA               NA
 2.857143        1       NA       NA           NA               NA
 3.000000        8       NA       NA           NA               NA
 3.142857        6       NA       NA           NA               NA
 3.285714        9       NA       NA           NA               NA
 3.428571       19       NA       NA           NA               NA
 3.571429       18       NA       NA           NA               NA
 3.714286       26       NA       NA           NA               NA
 [ reached getOption("max.print") -- omitted 399 rows ]

c) Con un gráfico mida el error de cada uno de los modelos anteriores y determine cual de los modelos es el mejor.

El mejor modelo es Holt -Winters Calibrado, debido a que tiene mayor correlación y el menor error de todos.

d) Con el mejor modelo encontrado en el punto anterior genere la predicción de un mes, pero esta vez utilizando toda la serie de tiempo. Grafique la serie original y la predicción junto con el limite inferior y superior.

         Point Forecast     Lo 95     Hi 95
60.85714       1989.630 1658.9488  2320.312
61.00000       2047.884 1644.2364  2451.533
61.14286       2258.683 1736.8770  2780.490
61.28571       2825.057 2148.9767  3501.136
61.42857       3169.665 2311.1663  4028.164
61.57143       3385.451 2321.6016  4449.300
61.71429       3558.011 2269.2021  4846.821
61.85714       3845.539 2296.2690  5394.809
62.00000       3903.793 2098.9267  5708.659
62.14286       4114.592 2038.9239  6190.260
62.28571       4680.965 2320.3536  7041.576
62.42857       5025.574 2366.7224  7684.425
62.57143       5241.359 2271.6632  8211.055
62.71429       5413.920 2121.3558  8706.484
62.85714       5701.447 2060.4963  9342.398
63.00000       5759.702 1774.4760  9744.927
63.14286       5970.501 1630.1258 10310.875
63.28571       6536.874 1830.8459 11242.901
63.42857       6881.482 1799.6286 11963.336
63.57143       7097.268 1629.7126 12564.823
63.71429       7269.829 1406.9673 13132.690
63.85714       7557.356 1278.0649 13836.647
64.00000       7615.610  923.2521 14307.968
64.14286       7826.409  711.9922 14940.826
64.28571       8392.782  847.5190 15938.045
 [ reached 'max' / getOption("max.print") -- omitted 5 rows ]
Time Series:
Start = c(1, 6) 
End = c(65, 1) 
Frequency = 7 
          serie p.prediccion   p.LimInf  p.LimSup
 1.714286     2           NA         NA        NA
 1.857143     5           NA         NA        NA
 2.000000     3           NA         NA        NA
 2.142857     2           NA         NA        NA
 2.285714     1           NA         NA        NA
 2.428571     9           NA         NA        NA
 2.571429     1           NA         NA        NA
 2.714286     3           NA         NA        NA
 2.857143     1           NA         NA        NA
 3.000000     8           NA         NA        NA
 3.142857     6           NA         NA        NA
 3.285714     9           NA         NA        NA
 3.428571    19           NA         NA        NA
 3.571429    18           NA         NA        NA
 3.714286    26           NA         NA        NA
 3.857143     4           NA         NA        NA
 4.000000    17           NA         NA        NA
 4.142857    24           NA         NA        NA
 [ reached getOption("max.print") -- omitted 426 rows ]

2. [35 puntos] La tabla trafico trafico_tren_V2.csv contiene la cantidad de personas que se suben al metro por hora. Tomando los datos a partir del 15 de Enero del 2016 realice lo siguiente:

(a) Encuentre las fechas faltantes y con un suavizado de 25 impute valor a dichas fechas.

(b) Convierta a serie de tiempo (utilice Patrón-estacional diario).

Time Series:
Start = c(1, 1) 
End = c(990, 24) 
Frequency = 24 
 [1]  753.000  481.000 3515.375 3515.375  829.000 3845.111 5394.000 3940.500
 [9] 5502.000 3849.727 4447.000 3643.923 5154.000 3508.429 5563.000 3526.786
[17] 6483.000 3516.714 4799.000 3128.067 2942.000 2934.800 3324.000 1700.000
[25] 1467.000 1033.000  738.000 2877.750 2981.176  688.000 1211.000 1870.000
[33] 2603.000 2691.000 3746.000 4207.000 4601.000 2628.812 4632.000 2579.812
[41] 4636.000 2455.471 3917.000 2482.529 2683.000 2484.824 2349.000 2082.000
[49] 1065.000 2333.588  644.000  343.000  466.000 2229.294  929.000 1430.000
[57] 1988.000 2524.000 2857.000 3172.000 2006.059 3954.000 3746.000 1940.706
[65] 3749.000 2144.188 2891.000 2457.059 2084.000 2679.882 1284.000  977.000
[73] 2816.062  335.000  263.000
 [ reached getOption("max.print") -- omitted 23685 entries ]

(c) Usando las últimas 24 horas para pruebas y el resto de fechas para entrenamiento genere los modelos de HOLT-WINTERS, HOLT-WINTERS Calibrado y Redes Neuronales, luego en un solo gráfico muestre la serie de entrenamiento, la serie de prueba y el resultado de la predicción de cada uno de los modelos anteriores.

HOLT-Winter

         Point Forecast     Lo 80    Hi 80      Lo 95     Hi 95
990.0000       2793.276 1922.0662 3664.486 1460.87565  4125.676
990.0417       2474.484 1357.4468 3591.521  766.12303  4182.845
990.0833       2432.369 1114.5446 3750.193  416.93056  4447.807
990.1250       2357.611  865.7421 3849.479   75.99452  4639.227
990.1667       2714.851 1067.1839 4362.518  194.96162  5234.740
990.2083       4202.956 2412.9655 5992.945 1465.40210  6940.509
990.2500       6156.367 4234.5330 8078.200 3217.17573  9095.558
990.2917       6760.799 4715.5735 8806.025 3632.89647  9888.702
990.3333       6560.519 4398.9052 8722.133 3254.61593  9866.422
990.3750       6328.135 4056.0602 8600.210 2853.29647  9802.974
990.4167       6117.080 3739.6447 8494.514 2481.10674  9753.052
990.4583       6352.817 3874.4720 8831.161 2562.51564 10143.118
990.5000       6557.475 3982.1477 9132.802 2618.85178 10496.098
990.5417       6532.777 3863.9659 9201.588 2451.18263 10614.371
990.5833       6720.864 3961.7129 9480.016 2501.10634 10940.623
 [ reached 'max' / getOption("max.print") -- omitted 9 rows ]
         Point Forecast     Lo 80    Hi 80      Lo 95     Hi 95
990.0000       2793.276 1922.0662 3664.486 1460.87565  4125.676
990.0417       2474.484 1357.4468 3591.521  766.12303  4182.845
990.0833       2432.369 1114.5446 3750.193  416.93056  4447.807
990.1250       2357.611  865.7421 3849.479   75.99452  4639.227
990.1667       2714.851 1067.1839 4362.518  194.96162  5234.740
990.2083       4202.956 2412.9655 5992.945 1465.40210  6940.509
990.2500       6156.367 4234.5330 8078.200 3217.17573  9095.558
990.2917       6760.799 4715.5735 8806.025 3632.89647  9888.702
990.3333       6560.519 4398.9052 8722.133 3254.61593  9866.422
990.3750       6328.135 4056.0602 8600.210 2853.29647  9802.974
990.4167       6117.080 3739.6447 8494.514 2481.10674  9753.052
990.4583       6352.817 3874.4720 8831.161 2562.51564 10143.118
990.5000       6557.475 3982.1477 9132.802 2618.85178 10496.098
990.5417       6532.777 3863.9659 9201.588 2451.18263 10614.371
990.5833       6720.864 3961.7129 9480.016 2501.10634 10940.623
 [ reached 'max' / getOption("max.print") -- omitted 9 rows ]

calibrado HW

Holt-Winters exponential smoothing with trend and additive seasonal component.

Call:
HoltWinters(x = serie.aprendizaje, alpha = alpha.i, beta = beta.i,     gamma = gamma.i)

Smoothing parameters:
 alpha: 0.1
 beta : 0
 gamma: 1

Coefficients:
             [,1]
a   -302752.86271
b       -47.29718
s1   303284.38627
s2   302434.46961
s3   302123.59654
s4   302010.45111
s5   302167.58107
s6   302727.16609
s7   303778.80105
s8   304871.03307
s9   306120.50073
s10  306803.28144
s11  307110.52675
s12  307402.16449
s13  307648.60570
s14  307509.70405
s15  307565.80321
s16  307576.26255
s17  307827.64542
s18  307983.63477
s19  307653.39360
s20  306869.91626
s21  306330.01066
s22  305878.75462
s23  305475.17219
s24  306608.86271
         Point Forecast      Lo 80     Hi 80      Lo 95    Hi 95
990.0000       484.2264  -890.1564 1858.6092 -1617.7107 2586.163
990.0417      -412.9875 -1794.2251  968.2502 -2525.4081 1699.433
990.0833      -771.1577 -2159.2163  616.9009 -2894.0101 1351.695
990.1250      -931.6003 -2326.4465  463.2459 -3064.8335 1201.633
990.1667      -821.7675 -2223.3685  579.8334 -2965.3312 1321.796
990.2083      -309.4797 -1717.8030 1098.8436 -2463.3243 1844.365
990.2500       694.8581  -720.1556 2109.8718 -1469.2186 2858.935
990.2917      1739.7929   318.1203 3161.4655  -434.4677 3914.054
990.3333      2941.9634  1513.6629 4370.2639   757.5663 5126.361
990.3750      3577.4469  2142.5492 5012.3447  1382.9602 5771.934
990.4167      3837.3951  2395.9302 5278.8599  1632.8648 6041.925
990.4583      4081.7356  2633.7335 5529.7378  1867.2075 6296.264
990.5000      4280.8797  2826.3696 5735.3897  2056.3985 6505.361
990.5417      4094.6808  2633.6919 5555.6698  1860.2910 6329.071
990.5833      4103.4828  2636.0435 5570.9221  1859.2281 6347.737
 [ reached 'max' / getOption("max.print") -- omitted 9 rows ]

Redes Neuronales

Series: train.ts 
Model:  NNAR(43,1,10)[24] 
Call:   nnetar(y = train.ts, size = 10)

Average of 20 networks, each of which is
a 43-10-1 network with 451 weights
options were - linear output units 

sigma^2 estimated as 189260
       Point Forecast
990.00      2055.2810
990.04      1463.2901
990.08       852.0951
990.12       609.3308
990.17       427.2401
990.21       414.5691
990.25       751.5265
990.29      1165.6261
990.33      1881.2204
990.38      2499.5063
990.42      3207.6977
990.46      3799.0823
990.50      4044.5884
990.54      4134.4537
990.58      4334.4183
990.62      4308.7862
990.67      4277.5442
990.71      4285.0110
990.75      4104.9075
990.79      3768.1915
990.83      3194.2490
990.88      2778.3316
990.92      2279.5560
990.96      2071.2922
Time Series:
Start = c(1, 1) 
End = c(990, 24) 
Frequency = 24 
           train.ts Original       HW HW Calibrado Redes neuronales
  1.000000  753.000       NA       NA           NA               NA
  1.041667  481.000       NA       NA           NA               NA
  1.083333 3515.375       NA       NA           NA               NA
  1.125000 3515.375       NA       NA           NA               NA
  1.166667  829.000       NA       NA           NA               NA
  1.208333 3845.111       NA       NA           NA               NA
  1.250000 5394.000       NA       NA           NA               NA
  1.291667 3940.500       NA       NA           NA               NA
  1.333333 5502.000       NA       NA           NA               NA
  1.375000 3849.727       NA       NA           NA               NA
  1.416667 4447.000       NA       NA           NA               NA
  1.458333 3643.923       NA       NA           NA               NA
  1.500000 5154.000       NA       NA           NA               NA
  1.541667 3508.429       NA       NA           NA               NA
  1.583333 5563.000       NA       NA           NA               NA
 [ reached getOption("max.print") -- omitted 23745 rows ]

(e) Con el mejor modelo encontrado en el punto anterior genere la predicción de un día, pero esta vez utilizando toda la serie de tiempo. Grafique la serie original y la predicción junto con el límite inferior y superior con dygraph.

3. [30 puntos] Para la tabla Cajero.csv visto en clase, genere las reglas para la predicción del retiro de dinero de los días 14, 15 y 16 de agosto del 2012. Puede usar la regla que se implementó en el ejemplo de la clase para el día 15 de agosto del 2012.

14 de agosto

numeric(0)
Time Series:
Start = c(1998, 1) 
End = c(1998, 6) 
Frequency = 365 
[1]  628000 1162000 3891000 3005000 2716000 2621000
     fit 
11572558 
[1] 16657000
[1] -5084442
[1] 1.439353
     fit 
16657000 
[1] 16657000
Time Series:
Start = c(1998, 1) 
End = c(2012, 211) 
Frequency = 365 
 [1]   628000  1162000  3891000  3005000  2716000  2621000  3293000  2794000
 [9]   866000  2701000  2574000  2627000  3962000 11005000  1759000  2398000
[17]  5250000  3831000  3505000  3700000  3968000  2184000  1691000  2542000
[25]  2906000  2573000  4000000  6721000  4647000  2247000  7414000  5029000
[33]  3818000  3909000  4001000  3808000  1819000  3931000  2391000  2393000
[41]  2655000  4079000  3170000  1685000  4169000  7738000  6172000  3814000
[49]  3213000  2923000   985000  1567000  2403000  2737000  2757000  4695000
[57]  2584000  1376000  4354000 10641000  5647000  4938000  3869000  2513000
[65]  1074000  3463000  2389000  2616000  3201000  4733000  2194000  1153000
[73]  4310000  4764000  7648000
 [ reached getOption("max.print") -- omitted 5246 entries ]
[1] 14539697
[1] 20927760

15 de agosto

Time Series:
Start = c(1998, 1) 
End = c(1998, 6) 
Frequency = 365 
[1]  628000 1162000 3891000 3005000 2716000 2621000
     fit 
13543013 
[1] 15625000
[1] -2081987
[1] 1.153731
     fit 
15625000 
[1] 15625000
Time Series:
Start = c(1998, 1) 
End = c(2012, 211) 
Frequency = 365 
 [1]   628000  1162000  3891000  3005000  2716000  2621000  3293000  2794000
 [9]   866000  2701000  2574000  2627000  3962000 11005000  1759000  2398000
[17]  5250000  3831000  3505000  3700000  3968000  2184000  1691000  2542000
[25]  2906000  2573000  4000000  6721000  4647000  2247000  7414000  5029000
[33]  3818000  3909000  4001000  3808000  1819000  3931000  2391000  2393000
[41]  2655000  4079000  3170000  1685000  4169000  7738000  6172000  3814000
[49]  3213000  2923000   985000  1567000  2403000  2737000  2757000  4695000
[57]  2584000  1376000  4354000 10641000  5647000  4938000  3869000  2513000
[65]  1074000  3463000  2389000  2616000  3201000  4733000  2194000  1153000
[73]  4310000  4764000  7648000
 [ reached getOption("max.print") -- omitted 5246 entries ]
[1] 15216130
[1] 17555328

16 de agosto

Time Series:
Start = c(1998, 1) 
End = c(1998, 6) 
Frequency = 365 
[1]  628000 1162000 3891000 3005000 2716000 2621000
     fit 
13062527 
[1] 9269000
[1] 3793527
[1] 0.709587
    fit 
9269000 
[1] 9269000
Time Series:
Start = c(1998, 1) 
End = c(2012, 211) 
Frequency = 365 
 [1]   628000  1162000  3891000  3005000  2716000  2621000  3293000  2794000
 [9]   866000  2701000  2574000  2627000  3962000 11005000  1759000  2398000
[17]  5250000  3831000  3505000  3700000  3968000  2184000  1691000  2542000
[25]  2906000  2573000  4000000  6721000  4647000  2247000  7414000  5029000
[33]  3818000  3909000  4001000  3808000  1819000  3931000  2391000  2393000
[41]  2655000  4079000  3170000  1685000  4169000  7738000  6172000  3814000
[49]  3213000  2923000   985000  1567000  2403000  2737000  2757000  4695000
[57]  2584000  1376000  4354000 10641000  5647000  4938000  3869000  2513000
[65]  1074000  3463000  2389000  2616000  3201000  4733000  2194000  1153000
[73]  4310000  4764000  7648000
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[1] 12642804
[1] 8971169

4. [20 puntos] Usando el Método Multiplicativo de Holt-Winters que se presenta en las siguientes ecuaciones:

\[\begin{align*} \hat{x}_{t+h|t} &= (l_t + hb_t ) \cdot s_{t-m+h^{+}_{m}} \\ l_t &= \alpha \dfrac{y_t}{s_{t-m}} + (1-\alpha)(l_{t-1} + b_{t-1}) \\ b_t &= \beta (l_t -l_{t-1}) + (1-\beta)b_{t-1} \\ s_t &= \gamma \dfrac{y_t}{(l_{t-1}+b_{t-1})}+(1-\gamma)s_{t-m} \end{align*}\]

Programe una función recursiva que permita verificar los datos que se muestran en la siguiente tabla, para esto utilice los datos de visitas de turistas a Australia. Además tome \(\alpha = 0,441\), \(\beta = 0,030\) y \(\gamma = 0,002\) (recuerde que estos parámetros se obtienen minimizado el RMSE), además tome \(l_0 = 32,49\), \(b_0 = 0,70\) y \(s_0 = 1,02\):

5. [20 puntos] (Suavizado exponencial simple) Se tienen \(x_1 , ... , x_n\) y se desea pronosticar (predecir) \(x_{n+h}\) con \(h \in \mathbb{N}\) y donde \(\hat{x}_{n,h}\) denota la predicción de \(x_{n+h}\). Para \(\alpha \in ]0;1[\) , se define la

predicción por suavizado exponencial simple como sigue:

\[ \hat{x}_{n,h} = \alpha \sum_{j=0}^{n-1}(1-\alpha)^{j} x_{n-j} \] Lo anterior se puede calcular recursivamente usando la siguiente fórmula:

\[ \hat{x}_{n,h} \alpha x_n + (1-\alpha)\hat{x}_{n-1,h} \] Pruebe que la predicción \(\hat{x}_{n,h}\) es asintóticamente la solución de:

\[ \hat{x}_{n,h} = arg_{x} mín \sum_{j=0}^{n-1} (1-\alpha)^j (x_{n-j} - x)^2 \]

Solución: